Abstract
In this article, a shifted Legendre tau method is introduced to get a direct solution technique for solving multi-order fractional differential equations (FDEs) with constant coefficients subject to multi-point boundary conditions. The fractional derivative is described in the Caputo sense. Also, this article reports a systematic quadrature tau method for numerically solving multi-point boundary value problems of fractional-order with variable coefficients. Here the approximation is based on shifted Legendre polynomials and the quadrature rule is treated on shifted Legendre Gauss-Lobatto points. We also present a Gauss-Lobatto shifted Legendre collocation method for solving nonlinear multi-order FDEs with multi-point boundary conditions. The main characteristic behind this approach is that it reduces such problem to those of solving a system of algebraic equations. Thus we can find directly the spectral solution of the proposed problem. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms.
Highlights
Fractional calculus, as generalization of integer order integration and differentiation to its non-integer order counterpart, has proved to be a valuable tool in the modeling of many phenomena in the fields of physics, chemistry, engineering, aerodynamics, electrodynamics of complex medium, polymer rheology, etc. [1,2,3,4,5,6,7,8,9]
We focus on providing a numerical scheme, based on spectral methods, to solve multi-point boundary conditions for linear and nonlinear fractional differential equations (FDEs)
Regarding problem (43) subject to the three types of multi-point boundary conditions (44)-(46), we study two different cases of a, b, c, ν 1, ν2, and ν
Summary
Fractional calculus, as generalization of integer order integration and differentiation to its non-integer (fractional) order counterpart, has proved to be a valuable tool in the modeling of many phenomena in the fields of physics, chemistry, engineering, aerodynamics, electrodynamics of complex medium, polymer rheology, etc. [1,2,3,4,5,6,7,8,9]. The treatment of the nonlinear multi-order fractional multi-point value problems; with leading fractional-differential operator of order ν (m - 1
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